Math Formulas/Equations
What You Need to Know
On SAT Math, a huge share of points comes from recognizing which equation/formula models the situation and then manipulating it cleanly. This topic is about your core toolbox:
Algebra rules (distributing, factoring, exponents/radicals)
Solving equations/inequalities (linear, quadratic, rational, absolute value)
Core function models (linear, exponential)
Coordinate geometry equations (line formulas, circle equation)
If you can (1) rewrite expressions correctly and (2) solve for the variable without illegal moves, you’ll crush most “equation” questions.
Critical reminder: Whatever you do to one side of an equation/inequality, do it to the other side too. The only time the “same operation” rule changes is inequalities: multiplying/dividing by a negative flips the sign.
Core idea
An equation states two expressions are equal and you’re finding values that make it true. An inequality compares expressions with , \ge and you’re finding values that make it true.
Step-by-Step Breakdown
A) Solving linear equations (one variable)
Simplify each side: distribute, combine like terms.
Get variables on one side (add/subtract terms).
Isolate the variable (multiply/divide).
Check for special cases:
No solution: variable cancels and you get false, e.g. 0 = 5.
Infinitely many solutions: you get true, e.g. 0 = 0.
Mini-example: Solve 3(x-2)=2x+5
Distribute: 3x-6=2x+5
Subtract 2x: x-6=5
Add 6: x=11
B) Solving linear inequalities
Solve like an equation.
Flip the inequality only if you multiply/divide by a negative.
Write in interval form if asked.
Mini-example: Solve -2x+1 \le 7
Subtract 1: -2x \le 6
Divide by -2 (flip!): x \ge -3
SAT trap: Forgetting to flip the sign when dividing by a negative.
C) Solving systems of linear equations
Use substitution or elimination.
Elimination steps:
Align equations as ax+by=c format.
Multiply one/both equations so a variable’s coefficients are opposites.
Add/subtract to eliminate.
Solve for remaining variable.
Back-substitute.
Mini-example:
Solve
x+y=9
x-y=1
Add: 2x=10 \Rightarrow x=5, then y=4.
D) Solving quadratics
Quadratics appear as ax^2+bx+c=0.
Method choice (fast decision):
If it factors nicely, factor.
If it’s already a(x-h)^2+k, use vertex form.
If not factorable, use the quadratic formula.
Factoring steps:
Move everything to one side: =0.
Factor.
Set each factor to zero.
Mini-example: x^2-5x+6=0
Factor: (x-2)(x-3)=0 so x=2 or x=3.
E) Rational equations (variables in denominators)
State restrictions: denominator \ne 0.
Multiply both sides by the LCD (least common denominator).
Solve the resulting equation.
Check for extraneous solutions (plug back).
Mini-example: Solve \frac{2}{x-1}=3
Restriction: x \ne 1
Multiply: 2=3(x-1)=3x-3
3x=5 \Rightarrow x=\frac{5}{3} (valid)
F) Absolute value equations/inequalities
Equation rule:
|A|=b (with b\ge 0) becomes A=b **or** A=-b.
Inequality rules:
|A|
|A|>b becomes A>b **or** A<-b
Mini-example: |2x-1|=5
2x-1=5 \Rightarrow x=3
2x-1=-5 \Rightarrow x=-2
G) Exponential equations (SAT-level)
Often solvable by rewriting to a common base.
Mini-example: Solve 2^{x+1}=8
Rewrite 8=2^3, so x+1=3 \Rightarrow x=2.
Key Formulas, Rules & Facts
Algebra + equation-solving essentials
Formula / Rule | When to use | Notes / pitfalls |
|---|---|---|
a(b+c)=ab+ac | Distribute | Don’t forget to distribute to every term |
ab+ac=a(b+c) | Factor (GCF) | Always check for a GCF first |
(x+m)(x+n)=x^2+(m+n)x+mn | Multiply binomials | Useful to reverse when factoring |
x^2-y^2=(x-y)(x+y) | Difference of squares | Common fast factor on SAT |
(x\pm y)^2=x^2\pm 2xy+y^2 | Perfect square trinomials | Spot patterns for fast factoring |
If ab=0, then a=0 or b=0 | Zero product property | Only works when product equals 0 |
\frac{a}{b}=\frac{c}{d} \Rightarrow ad=bc | Proportions | Ensure b\ne 0 and d\ne 0 |
Exponents & radicals
Rule | When to use | Notes / pitfalls |
|---|---|---|
a^m\cdot a^n=a^{m+n} | Multiply same base | Bases must match |
\frac{a^m}{a^n}=a^{m-n} | Divide same base | If m |
(a^m)^n=a^{mn} | Power of a power | Common place to slip |
(ab)^n=a^n b^n | Distribute exponent | Works for multiplication |
\left(\frac{a}{b}\right)^n=\frac{a^n}{b^n} | Power of a fraction | Need b\ne 0 |
a^{-n}=\frac{1}{a^n} | Negative exponents | Means “reciprocal” |
\sqrt{a}\sqrt{b}=\sqrt{ab} (for a,b\ge 0) | Multiply radicals | Don’t combine across plus: \sqrt{a}+\sqrt{b} stays |
\sqrt{a^2}=|a| | Simplifying radicals | SAT may test absolute value nuance |
Linear functions & lines
Formula | When to use | Notes |
|---|---|---|
m=\frac{y_2-y_1}{x_2-x_1} | Slope between two points | Vertical line: x_2=x_1 (undefined slope) |
y=mx+b | Line form | b is y-intercept |
y-y_1=m(x-x_1) | Point-slope form | Great when you have slope + a point |
Standard form Ax+By=C | Systems / intercepts | Slope is -\frac{A}{B} (if B\ne 0) |
Parallel lines: m_1=m_2 | Identify parallel | Same slope |
Perpendicular: m_1m_2=-1 | Identify perpendicular | Negative reciprocal slopes |
Coordinate geometry
Formula | When to use | Notes |
|---|---|---|
Distance: d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2} | Segment length | Comes from Pythagorean theorem |
Midpoint: \left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right) | Midpoint of segment | Average the coordinates |
Circle: (x-h)^2+(y-k)^2=r^2 | Circle graph/equation | Center (h,k), radius r |
Quadratics (must-know relationships)
Formula / fact | When to use | Notes |
|---|---|---|
Quadratic formula: x=\frac{-b\pm\sqrt{b^2-4ac}}{2a} | Solve any quadratic | Watch sign of b carefully |
Discriminant: \Delta=b^2-4ac | Number of real solutions | \Delta>0 two, \Delta=0 one, \Delta<0 none (real) |
Vertex: for ax^2+bx+c, x=\frac{-b}{2a} | Find axis of symmetry | Then plug in for y |
Vertex form: a(x-h)^2+k | Graph transformations | Vertex at (h,k) |
Percent, rate, and growth equations
Equation | When to use | Notes |
|---|---|---|
Percent change: \frac{\text{new}-\text{old}}{\text{old}}\times 100\% | Increase/decrease | “Of” usually means multiply |
Simple interest: I=Prt | Interest problems | r as decimal, t in years |
Exponential growth/decay: A=A_0(1\pm r)^t | Repeated percent change | Use -$ for decay, +$ for growth |
Average speed: v=\frac{d}{t} | Motion problems | Total avg speed is \frac{\text{total }d}{\text{total }t} |
Examples & Applications
1) Rearranging a formula (literal equations)
Problem style: Solve for a variable in a given formula.
Given V=\frac{1}{3}\pi r^2 h, solve for h.
Multiply both sides by 3: 3V=\pi r^2 h
Divide by \pi r^2: h=\frac{3V}{\pi r^2}
Key insight: Treat it like isolating x—just keep operations balanced.
2) System from a word problem
A theater sold 100 tickets. Adult tickets cost \$12 and student tickets cost \$8. Total revenue was \$1040. How many student tickets?
Let a = adult, s = student.
Count: a+s=100
Revenue: 12a+8s=1040
Eliminate: multiply first equation by 8:
8a+8s=800
Subtract from revenue equation:
(12a+8s)-(8a+8s)=1040-800 \Rightarrow 4a=240 \Rightarrow a=60
So s=40.
Key insight: “Total number” + “total value” is almost always a 2-equation system.
3) Quadratic from geometry (area)
A rectangle has area 48 and length x+2 and width x-2. Find x.
Set up: (x+2)(x-2)=48
Use difference of squares: x^2-4=48
So x^2=52 \Rightarrow x=\pm\sqrt{52}=\pm 2\sqrt{13}.
SAT reality check: If x is a dimension parameter, you may need **positive only** (and also ensure x-2>0). So x=2\sqrt{13} works.
4) Inequality + interval solution
Solve and graph: 2(1-x)>x+4
Distribute: 2-2x>x+4
Subtract 2: -2x>x+2
Subtract x: -3x>2
Divide by -3 (flip): x< -\frac{2}{3}
Key insight: The only “special move” is flipping the sign when dividing by a negative.
Common Mistakes & Traps
Forgetting to distribute a negative
Wrong: -(x-3)=-x-3
Right: -(x-3)=-x+3
Fix: Treat -1 as the multiplier and distribute carefully.
Combining unlike terms
Wrong: 2x+3=5x
Why wrong: 3 isn’t an x-term.
Fix: Only combine terms with the exact same variable part.
Illegal canceling across addition
Wrong: \frac{x+2}{x}=\frac{2}{1}
Why wrong: You can only cancel factors, not terms in a sum.
Fix: Factor first if possible.
Not flipping the inequality when dividing by a negative
Wrong: -2x<6 \Rightarrow x< -3
Right: -2x -3
Fix: Circle the sign whenever you divide/multiply by a negative.
Dropping solutions when solving quadratics
Wrong: Taking only the + in \pm or missing the second factor.
Fix: For x^2=9, write x=\pm 3 unless context restricts.
Extraneous solutions in rational/absolute value equations
What happens: Multiplying by a variable expression can introduce invalid answers.
Fix: State restrictions (like x\ne 1) and plug solutions back.
Misreading function notation
Mistake: Thinking f(x+2) equals f(x)+2.
Fix: Replace the entire input: if f(x)=x^2, then f(x+2)=(x+2)^2.
Sign errors in the quadratic formula
Common slip: Using -b incorrectly when b is negative.
Fix: If b=-5, then -b=5. Put parentheses: -(-5)=5.
Memory Aids & Quick Tricks
Trick / mnemonic | What it helps you remember | When to use |
|---|---|---|
“Same base, add/subtract exponents” | a^m a^n=a^{m+n} and a^m/a^n=a^{m-n} | Simplifying exponent expressions |
“SOH-CAH-TOA” | \sin=\frac{\text{opp}}{\text{hyp}} etc. | Only if trig appears (rare), but can help with right-triangle ratios |
“FOIL” | Multiply (a+b)(c+d) | Expanding binomials (though distributing is safer) |
“Flip when negative” | Inequality sign flips when dividing/multiplying by negative | Inequalities |
“Circle form = Center/Radius” | (x-h)^2+(y-k)^2=r^2 gives center (h,k) | Circle equation questions |
“Axis is -b/2a” | Quick vertex x-coordinate | Quadratic graphs/vertex/maximum-minimum |
Quick Review Checklist
You can distribute, combine like terms, and factor (GCF, difference of squares, perfect square patterns).
You know exponent rules, including a^{-n}=\frac{1}{a^n} and \sqrt{a^2}=|a|.
When solving equations, you isolate the variable and watch for no solution vs infinite solutions.
For inequalities, you flip the sign when multiplying/dividing by a negative.
For systems, you can do elimination quickly and interpret solutions.
For quadratics, you can factor or use x=\frac{-b\pm\sqrt{b^2-4ac}}{2a} and use \Delta=b^2-4ac to predict roots.
For rational equations, you state restrictions and check for extraneous answers.
You can use line and coordinate formulas: m=\frac{y_2-y_1}{x_2-x_1}, d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}, midpoint, and (x-h)^2+(y-k)^2=r^2.
You don’t need new tricks—just clean algebra and careful sign handling.